Safe prime

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.
View other properties of prime numbers | View other properties of natural numbers

Definition

A prime number $p$ is termed a safe prime if $p$ is odd and $(p-1)/2$ is also a prime number.

The corresponding prime $(p-1)/2$ is termed a Sophie Germain prime.

Occurrence

Initial values

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A005385

The first few safe primes are:

5, 7, 11, 23, 47, 59, 83, 107, 167, [SHOW MORE]

179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903

View list on OEIS

The first few primes that are not safe primes are: 2, 3, 13, 17, 19.

Density in primes

Cutoff $n$	Number of primes $\leq n$	Number of safe primes $\leq n$	Proportion of primes that are safe primes
10	4	2	$1/2=0.5$
100	25	7	$7/25=0.28$
1000	168	24	$25/168\approx 0.15$

Relation with other properties

Related properties of primes or pairs of primes

• Sophie Germain prime is a prime $q$ such that $2q+1$ is also a prime.
• Twin primes are primes that differ by two.

Related properties of longer chains of primes

• Cunningham chain of the first kind is a chain of primes $q_{1}<q_{2}<\dots <q_{k}$ such that $q_{i+1}=2q_{i}+1$.
• Bitwin chain is a collection of multiple pairs of twin primes, each pair being double the previous one.

Facts

• Quadratic nonresidue that is not minus one is primitive root for safe prime
• Safe prime has plus or minus two as a primitive root